# Tagged Questions

Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).

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### The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy type of the semi-algebraic set defined by $$P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r .$$ Is there a ...
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### Hypersurfaces without real points

Let $n, d$ be positive integers. I am interested in the open subset $\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to ...
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### Cohomology of Structure Sheaves: Algebraic, Constructible and more

I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
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### Isolated zero set of real polynomial in two variables

The polynomial $x^2+y^2$ has an isolated zero at the origin. And so do powers $(x^2+y^2)^n$ of this polynomial. I'm wondering if this is a special property of these real polynomials. Here's the ...
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### Conditions on coefficients of a homogeneous polynomial of the third degree in three variables over R which allow it to be positive on a positive octant

I am a student of Saint Petersburg State Polytechnical University, chair of Theoretical Mechanics. While looking into stability of ideal crystal lattices (2D and 3D) by means of molecular dynamics I ...
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Sturm's theorem gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P_1,\dotsc,P_n\in \... 1answer 439 views ### Function Fields of Real Varieties Let$V$be a geometrically irreducible and reduced scheme defined over the real numbers, and let$K = K(V)$be its function field. If$V$does not have any real points, is it true that$K$is not ... 5answers 2k views ### Polynomial positive on an interval If$p$is a polynomial with real coefficients and p(x)>0 on [0,1], then$p(x)=\sum c_{i,j} x^i(1-x)^j$with$c_{i,j}$positive. I know this is true but but I need a proof/reference. Thanks! 3answers 479 views ### Effective algorithm to test positivity Let$f(x_1,\ldots, x_n)$be a real polynomial in several variables. Is there an effective algorithm to test whether$f$is positive (or nonnegative) on the whole of${\mathbb{R}}^n$? 2answers 802 views ### Embeddings and triangulations of real analytic varieties This is a follow up question to my answer here How do you define the Euler Characteristic of a scheme? A real analytic space is a ringed space locally isomorphic to$(X,O/I)$where$X$is the zero ... 4answers 1k views ### Measure on real Grassmannians OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure$\theta$on$O(n)$, fix ... 4answers 3k views ### Real algebraic geometry vs. algebraic geometry This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with ... 2answers 398 views ### Real spectrum of ring of continuous semialgebraic functions Let R be a real closed field, and let U be a semialgebraic subset of$R^n$. Let$S^0(U)$be the ring of continuous R-valued semialgebraic functions. Also let$\tilde{U}$be the subset of Spec$_r (R[...
I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...